| L(s) = 1 | + (1.89 − 0.632i)2-s + (3.19 − 2.40i)4-s − 5.79·5-s + 5.87i·7-s + (4.55 − 6.57i)8-s + (−10.9 + 3.66i)10-s + 8.07i·11-s − 14.0·13-s + (3.71 + 11.1i)14-s + (4.47 − 15.3i)16-s − 29.9·17-s + 4.35i·19-s + (−18.5 + 13.9i)20-s + (5.11 + 15.3i)22-s + 8.74i·23-s + ⋯ |
| L(s) = 1 | + (0.948 − 0.316i)2-s + (0.799 − 0.600i)4-s − 1.15·5-s + 0.839i·7-s + (0.568 − 0.822i)8-s + (−1.09 + 0.366i)10-s + 0.734i·11-s − 1.07·13-s + (0.265 + 0.796i)14-s + (0.279 − 0.960i)16-s − 1.76·17-s + 0.229i·19-s + (−0.926 + 0.695i)20-s + (0.232 + 0.696i)22-s + 0.380i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7763582988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7763582988\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.89 + 0.632i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 + 5.79T + 25T^{2} \) |
| 7 | \( 1 - 5.87iT - 49T^{2} \) |
| 11 | \( 1 - 8.07iT - 121T^{2} \) |
| 13 | \( 1 + 14.0T + 169T^{2} \) |
| 17 | \( 1 + 29.9T + 289T^{2} \) |
| 23 | \( 1 - 8.74iT - 529T^{2} \) |
| 29 | \( 1 + 13.4T + 841T^{2} \) |
| 31 | \( 1 - 7.65iT - 961T^{2} \) |
| 37 | \( 1 - 25.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 49.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 47.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 14.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 100. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 34.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 66.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 40.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 87.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 28.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.35T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.90460754386351661717676597326, −9.843362871435117468652199501269, −8.929353411897756554128607779686, −7.71107378685949767871052709750, −7.04032624703013826174706930205, −5.99500693805081026800544529153, −4.79676865670567589319238363429, −4.27558918242613843529477858400, −2.98854177318842672738083661514, −1.98095072851468236612814515811,
0.18301970378896940961949315850, 2.37805401329468711648689146697, 3.66284279304861414067343593796, 4.30193143146240871483577303245, 5.20588417389213265377166237913, 6.58721087868663229100104930547, 7.17295082956104503633999266284, 7.981540790834291374131662232763, 8.816708087425752748690585452513, 10.27460907048141534860468974051
